Sooner and later waiting time problems for Bernoulli trials: Frequency and run quotas
In a sequence of i.i.d. Bernoulli trials the following waiting times are considered: the waiting time till the sooner of s successes in total or r failures in total occurs (frequency quota); the waiting time for a run of s successes or a run of r failures whichever comes sooner (run quota); and the corresponding waiting times for whichever comes later. The means and variances of these waiting times are obtained by deriving their generalized probability generating functions in the run quota case, and by using Dirichlet integrals of type 2 in the frequency quota case.
generalized probability generating functions
scientific article; zbMATH DE number 4138841